In nuclear oil well logging, a tool is lowered into an oil well and slowly withdrawn. During the removal process, nuclear particles are emitted into the surrounding formation and scattered radiation is detected as a function of depth. This recorded "log" is then interpreted into estimates of the formation's properties.
One such tool, for measuring the density and P.sub.e, photoelectric absorportion factor, of a formation surrounding the borehole, comprises a sonde body containing a gamma ray radioisotopic source and two gamma ray detectors. The detectors are typically NaI crystal scintillators which are spaced in the tool from the gamma ray source. "The Dual Spacing Formation Density Log", Journal of Petroleum Technology; Wahl, et al., December 1964 pp. 1411-1416, and U.S. Pat. Nos. 3,321,625 to Wahl, 3,864,569 to Tillman, and 4,048,495 to Ellis describe such a device in detail. These references describe the count rate of the far detector as an exponential function of the formation density. However, the mudcake in the borehole and borehole rugosity also affect the count rate of the far detector. For this reason, a second detector, the near detector, obtains information which is used to compensate for the effects of mudcake and rugosity on the far detector.
The Wahl article describes the compensation for the effects of mudcake and rugosity, which is known as the "spine-and-ribs" method. The spine-and-ribs method plots the short spacing detector counting rate for a particular energy window against the long spacing detector counting rate for a different energy window. The "spine", which has been developed through laboratory measurements, reflects the detector responses for variations in formation density only. The "ribs" extend from the spine and reflect the effect of mudcake thickness and mudcake density on the readings.
Thus, by using the count rates of the detectors as inputs, one finds a cross-plot location corresponding to a point on a rib extending from the spine. One then determines formation density by tracing the rib on which the point is located back to the spine, with the intersection of the rib and spine dictating the formation density, and the location on the rib dictating mudcake parameters.
Computer simulations are often used to predict the detector responses of such gamma-gamma density logging tools or other nuclear well logging tools. This routinely involves the use of 3-D Monte Carlo codes like MCNP (LASL Group X-6, "MCNP--A General Monte Carlo Code for Neutron and Photon Transport, Version 2B," LA-7396-M, Rev., Los Alamos National Lab., 1981) to solve the radiation transport equation under various combinations of mudcake and formation conditions. The major advantage of the Monte Carlo method (over other computational methods such as discrete ordinates) is that it can explicitly model the often asymmetric geometric details of both the tool and borehole environment. The major disadvantage is that the Monte Carlo method yields answers that are statistical in nature and therefore may require many cpu hours of computer time to achieve acceptable precision.
Modeling studies have typically involved estimating the response of a tool in a homogeneous formation. Under these conditions, a single calculation is sufficient to characterize the tool's response. If, on the other hand, the effects of borehole rugosity (i.e. borehole irregularities having a spatial scale which is smaller than the source-to-detector spacings) on density measurements were being studied, an estimated log of detector responses would be required. To generate this log with conventional Monte Carlo methods would require hundreds of cpu hours.
Creating a log of detector responses is computationally intensive because it involves a separate computer run for each tool position within the borehole. As the tool "moves up the borehole", the relative location of the tool with respect to the formation's heterogeneous features or borehole rugosity changes, and therefore a new model is needed for each vertical interval. A log can be simulated by interpolating between discrete stationary estimates of the detector responses. The inventors have used this approach to successfully predict the response of a neutron porosity tool past thin beds and isolated bed boundaries, but it required a large amount of computer time to simulate only a few thin bed cases. Thus, the need existed for the development of a more computationally efficient technique to make modeling a viable option for studying spatially inhomogeneous problems.